Learning Low-Degree Quantum Objects

Authors Srinivasan Arunachalam , Arkopal Dutt , Francisco Escudero Gutiérrez , Carlos Palazuelos

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Srinivasan Arunachalam
  • IBM Quantum, Thomas J Watson Research Center, Yorktown Heights, NY, USA
Arkopal Dutt
  • IBM Quantum, IBM Research Cambridge, MA, USA
Francisco Escudero Gutiérrez
  • CWI & QuSoft, Amsterdam, The Netherlands
Carlos Palazuelos
  • Dpto. Análisis Matemático y Matemática Aplicada, Fac. Ciencias Matemáticas, Universidad Complutense de Madrid, Spain
  • Instituto de Ciencias Matemáticas, Madrid, Spain


We thank the referees of ICALP`24 and TQC`24 for useful comments. We thank Jop Briët and Antonio Pérez Hernández for useful discussions. We thank Alexandros Eskenazis, Paata Ivanisvili, Alexander Volberg and Haonan Zhang for useful comments. A.D. and S.A. thank the Institute for Pure and Applied Mathematics (IPAM) for its hospitality throughout the long program "Mathematical and Computational Challenges in Quantum Computing" in Fall 2023 during which part of this work was initiated.

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Srinivasan Arunachalam, Arkopal Dutt, Francisco Escudero Gutiérrez, and Carlos Palazuelos. Learning Low-Degree Quantum Objects. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 13:1-13:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We consider the problem of learning low-degree quantum objects up to ε-error in 𝓁₂-distance. We show the following results: (i) unknown n-qubit degree-d (in the Pauli basis) quantum channels and unitaries can be learned using O(1/ε^d) queries (which is independent of n), (ii) polynomials p:{-1,1}ⁿ → [-1,1] arising from d-query quantum algorithms can be learned from O((1/ε)^d ⋅ log n) many random examples (x,p(x)) (which implies learnability even for d = O(log n)), and (iii) degree-d polynomials p:{-1,1}ⁿ → [-1,1] can be learned through O(1/ε^d) queries to a quantum unitary U_p that block-encodes p. Our main technical contributions are new Bohnenblust-Hille inequalities for quantum channels and completely bounded polynomials.

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  • Theory of computation → Quantum complexity theory
  • Tomography


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